On the entire functions from the Laguerre-Pólya class having the decreasing second quotients of Taylor coefficients

For an entire function f(z)=∑k=0∞akzk,ak>0, we show that f belongs to the Laguerre-Pólya class if the quotients an−12/an−2an are decreasing in n, and b:=limn→∞⁡an−12/an−2an such that b≥q∞(q∞≈3.2336). This estimation is sharp.